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Theorem ressusp 20636
Description: The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Hypotheses
Ref Expression
ressusp.1  |-  B  =  ( Base `  W
)
ressusp.2  |-  J  =  ( TopOpen `  W )
Assertion
Ref Expression
ressusp  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Ws  A
)  e. UnifSp )

Proof of Theorem ressusp
StepHypRef Expression
1 ressuss 20634 . . . . 5  |-  ( A  e.  J  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W
)t  ( A  X.  A
) ) )
213ad2ant3 1019 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  ( Ws  A ) )  =  ( (UnifSt `  W
)t  ( A  X.  A
) ) )
3 simp1 996 . . . . . . 7  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  W  e. UnifSp )
4 ressusp.1 . . . . . . . 8  |-  B  =  ( Base `  W
)
5 eqid 2467 . . . . . . . 8  |-  (UnifSt `  W )  =  (UnifSt `  W )
6 ressusp.2 . . . . . . . 8  |-  J  =  ( TopOpen `  W )
74, 5, 6isusp 20632 . . . . . . 7  |-  ( W  e. UnifSp 
<->  ( (UnifSt `  W
)  e.  (UnifOn `  B )  /\  J  =  (unifTop `  (UnifSt `  W
) ) ) )
83, 7sylib 196 . . . . . 6  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( (UnifSt `  W )  e.  (UnifOn `  B )  /\  J  =  (unifTop `  (UnifSt `  W
) ) ) )
98simpld 459 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  W
)  e.  (UnifOn `  B ) )
10 simp2 997 . . . . . . 7  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  W  e.  TopSp
)
114, 6istps 19306 . . . . . . 7  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  B ) )
1210, 11sylib 196 . . . . . 6  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  J  e.  (TopOn `  B ) )
13 simp3 998 . . . . . 6  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  e.  J )
14 toponss 19299 . . . . . 6  |-  ( ( J  e.  (TopOn `  B )  /\  A  e.  J )  ->  A  C_  B )
1512, 13, 14syl2anc 661 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  C_  B
)
16 trust 20600 . . . . 5  |-  ( ( (UnifSt `  W )  e.  (UnifOn `  B )  /\  A  C_  B )  ->  ( (UnifSt `  W )t  ( A  X.  A ) )  e.  (UnifOn `  A )
)
179, 15, 16syl2anc 661 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( (UnifSt `  W )t  ( A  X.  A ) )  e.  (UnifOn `  A )
)
182, 17eqeltrd 2555 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  ( Ws  A ) )  e.  (UnifOn `  A )
)
19 eqid 2467 . . . . . 6  |-  ( Ws  A )  =  ( Ws  A )
2019, 4ressbas2 14563 . . . . 5  |-  ( A 
C_  B  ->  A  =  ( Base `  ( Ws  A ) ) )
2115, 20syl 16 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  =  ( Base `  ( Ws  A
) ) )
2221fveq2d 5876 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifOn `  A
)  =  (UnifOn `  ( Base `  ( Ws  A
) ) ) )
2318, 22eleqtrd 2557 . 2  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (UnifSt `  ( Ws  A ) )  e.  (UnifOn `  ( Base `  ( Ws  A ) ) ) )
248simprd 463 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  J  =  (unifTop `  (UnifSt `  W
) ) )
2513, 24eleqtrd 2557 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  A  e.  (unifTop `  (UnifSt `  W
) ) )
26 restutopopn 20609 . . . 4  |-  ( ( (UnifSt `  W )  e.  (UnifOn `  B )  /\  A  e.  (unifTop `  (UnifSt `  W )
) )  ->  (
(unifTop `  (UnifSt `  W
) )t  A )  =  (unifTop `  ( (UnifSt `  W
)t  ( A  X.  A
) ) ) )
279, 25, 26syl2anc 661 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( (unifTop `  (UnifSt `  W )
)t 
A )  =  (unifTop `  ( (UnifSt `  W
)t  ( A  X.  A
) ) ) )
2824oveq1d 6310 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Jt  A
)  =  ( (unifTop `  (UnifSt `  W )
)t 
A ) )
292fveq2d 5876 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  (unifTop `  (UnifSt `  ( Ws  A ) ) )  =  (unifTop `  (
(UnifSt `  W )t  ( A  X.  A ) ) ) )
3027, 28, 293eqtr4d 2518 . 2  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Jt  A
)  =  (unifTop `  (UnifSt `  ( Ws  A ) ) ) )
31 eqid 2467 . . 3  |-  ( Base `  ( Ws  A ) )  =  ( Base `  ( Ws  A ) )
32 eqid 2467 . . 3  |-  (UnifSt `  ( Ws  A ) )  =  (UnifSt `  ( Ws  A
) )
3319, 6resstopn 19555 . . 3  |-  ( Jt  A )  =  ( TopOpen `  ( Ws  A ) )
3431, 32, 33isusp 20632 . 2  |-  ( ( Ws  A )  e. UnifSp  <->  ( (UnifSt `  ( Ws  A ) )  e.  (UnifOn `  ( Base `  ( Ws  A ) ) )  /\  ( Jt  A )  =  (unifTop `  (UnifSt `  ( Ws  A ) ) ) ) )
3523, 30, 34sylanbrc 664 1  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  A  e.  J
)  ->  ( Ws  A
)  e. UnifSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481    X. cxp 5003   ` cfv 5594  (class class class)co 6295   Basecbs 14507   ↾s cress 14508   ↾t crest 14693   TopOpenctopn 14694  TopOnctopon 19264   TopSpctps 19266  UnifOncust 20570  unifTopcutop 20601  UnifStcuss 20624  UnifSpcusp 20625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-tset 14591  df-unif 14595  df-rest 14695  df-topn 14696  df-top 19268  df-topon 19271  df-topsp 19272  df-ust 20571  df-utop 20602  df-uss 20627  df-usp 20628
This theorem is referenced by: (None)
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